MTGNexus

Am I reading Kalamax, the Stormsire and the possibilities correctly here?

I cast a sorcery, let's say Flame Jab while Kalmax is tapped.
I then cast Twincast on it.

Kalamax triggers and copies Twincast.
His ability lets me change targets for the copy, so I make the copy target the original Twincast.
This also triggers him getting a +1/+1 counter.

I let the copy resolve.
The Twincast copy makes a new Twincast copy, and lets me change the target.
I again make the new copy target the original Twincast.
My twincast copy copied an instant, so this also triggers for him to get another +1/+1 counter.

I let the new copy resolve.
The Twincast copy makes a new Twincast copy, and lets me change the target.
I again make the new copy target the original Twincast.
My twincast copy copied an instant, so this also triggers for him to get another +1/+1 counter.

..Rinse and repeat a 1,000 times to put 1,000 +1/+1 counters on Kalamax, the Stormsire.

SURELY I'm misreading something here?
Is going infinite as easy as casting Twincast, Fork, etc, as your first instant with a tapped Kalamax?

Yes, that works, as long as the copying instant is the first instant cast during a turn. You do need a valid target for the initial copying, but after that the rest happens as you say.